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On the geometry of the p-Laplacian operator

  • * Corresponding author: Bernd Kawohl

    * Corresponding author: Bernd Kawohl 
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  • The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $\Omega$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We report also on results concerning the normalized or game-theoretic $p$-Laplacian

    $\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$

    and its parabolic counterpart $u_t-\Delta_p^N u=0$. These equations are homogeneous of degree 1 and $\Delta_p^N$ is uniformly elliptic for any $p\in (1,\infty)$. In this respect it is more benign than the $p$-Laplacian, but it is not of divergence type.

    Mathematics Subject Classification: Primary: 35J92; Secondary: 35K92, 35D40, 49L25.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  The positive viscosity solution of (4.4)

    Figure 2.  Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc

    Figure 3.  Illustration of (5.4) and (5.5)

    Figure 4.  Numerical simulation of $u_{15}$ and side view in diagonal direction

    Figure 5.  Numerical simulation of $u_p$: normalized values along half of the diagonal for $p=2, 3, 4, 6, 8, 10, 15$ (left), and for $p=15$ compared to the line $y=x$ (right)

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